Average Error: 19.8 → 0.3
Time: 2.4m
Precision: 64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
↓
\[\frac{1}{\sqrt{x}} \cdot \frac{1}{(\left(\sqrt{x + 1}\right) \cdot \left(\sqrt{x}\right) + \left(x + 1\right))_*}\]
double f(double x) {
        double r19668045 = 1.0;
        double r19668046 = x;
        double r19668047 = sqrt(r19668046);
        double r19668048 = r19668045 / r19668047;
        double r19668049 = r19668046 + r19668045;
        double r19668050 = sqrt(r19668049);
        double r19668051 = r19668045 / r19668050;
        double r19668052 = r19668048 - r19668051;
        return r19668052;
}
↓
double f(double x) {
        double r19668053 = 1.0;
        double r19668054 = x;
        double r19668055 = sqrt(r19668054);
        double r19668056 = r19668053 / r19668055;
        double r19668057 = r19668054 + r19668053;
        double r19668058 = sqrt(r19668057);
        double r19668059 = fma(r19668058, r19668055, r19668057);
        double r19668060 = r19668053 / r19668059;
        double r19668061 = r19668056 * r19668060;
        return r19668061;
}


\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{1}{\sqrt{x}} \cdot \frac{1}{(\left(\sqrt{x + 1}\right) \cdot \left(\sqrt{x}\right) + \left(x + 1\right))_*}

Error

Bits error versus x

Target

Original19.8
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}}\]

Derivation

  1. Initial program 19.8

    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
  2. Using strategy rm

  3. Applied frac-sub19.8

    \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}}\]

  4. Simplified19.8

    \[\leadsto \frac{\color{blue}{\sqrt{x + 1} - \sqrt{x}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]
  5. Using strategy rm

  6. Applied flip--19.6

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x} \cdot \sqrt{x + 1}}\]

  7. Applied associate-/l/19.6

    \[\leadsto \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]

  8. Simplified0.8

    \[\leadsto \frac{\color{blue}{1}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]
  9. Using strategy rm

  10. Applied associate-/r*0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1} + \sqrt{x}}}\]
  11. Using strategy rm

  12. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\sqrt{x + 1} + \color{blue}{1 \cdot \sqrt{x}}}\]

  13. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\color{blue}{1 \cdot \sqrt{x + 1}} + 1 \cdot \sqrt{x}}\]

  14. Applied distribute-lft-out0.4

    \[\leadsto \frac{\frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}}{\color{blue}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}}\]

  15. Applied add-cube-cbrt0.4

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{x} \cdot \sqrt{x + 1}}}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]

  16. Applied times-frac0.4

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{x}} \cdot \frac{\sqrt[3]{1}}{\sqrt{x + 1}}}}{1 \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]

  17. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{x}}}{1} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{x + 1}}}{\sqrt{x + 1} + \sqrt{x}}}\]

  18. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{\sqrt{x}}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{x + 1}}}{\sqrt{x + 1} + \sqrt{x}}\]

  19. Simplified0.3

    \[\leadsto \frac{1}{\sqrt{x}} \cdot \color{blue}{\frac{1}{(\left(\sqrt{1 + x}\right) \cdot \left(\sqrt{x}\right) + \left(1 + x\right))_*}}\]

  20. Final simplification0.3

    \[\leadsto \frac{1}{\sqrt{x}} \cdot \frac{1}{(\left(\sqrt{x + 1}\right) \cdot \left(\sqrt{x}\right) + \left(x + 1\right))_*}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

  :herbie-target
  (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1)))))

  (- (/ 1 (sqrt x)) (/ 1 (sqrt (+ x 1)))))